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Farey sequence of order n is the sequence of completely reduced fractions, between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. Detailed explanation here.
Problem
The problem is, given n and k, where n = order of seq and k = element index, can we find the particular element from the sequence. For examples answer for (n=5, k =6) is 1/2.
Lead
There are many less than optimal solution available, but am looking for a near-optimal one. One such algorithm is discussed here, for which I am unable to understand the logic hence unable to apply the examples.
Question
Can some please explain the solution with more detail, preferably with an example.
Thank you.
I've read the method provided in your link, and the accepted C++ solution to it. Let me post them, for reference:
Editorial Explanation
Several less-than-optimal solutions exist. Using a priority queue, one
can iterate through the fractions (generating them one by one) in O(K
log N) time. Using a fancier math relation, this can be reduced to
O(K). However, neither of these solution obtains many points, because
the number of fractions (and thus K) is quadratic in N.
The “good” solution is based on meta-binary search. To construct this
solution, we need the following subroutine: given a fraction A/B
(which is not necessarily irreducible), find how many fractions from
the Farey sequence are less than this fraction. Suppose we had this
subroutine; then the algorithm works as follows:
Determine a number X such that the answer is between X/N and (X+1)/N; such a number can be determined by binary searching the range
1...N, thus calling the subroutine O(log N) times.
Make a list of all fractions A/B in the range X/N...(X+1)/N. For any given B, there is at most one A in this range, and it can be
determined trivially in O(1).
Determine the appropriate order statistic in this list (doing this in O(N log N) by sorting is good enough).
It remains to show how we can construct the desired subroutine. We
will show how it can be implemented in O(N log N), thus giving a O(N
log^2 N) algorithm overall. Let us denote by C[j] the number of
irreducible fractions i/j which are less than X/N. The algorithm is
based on the following observation: C[j] = floor(X*B/N) – Sum(C[D],
where D divides j). A direct implementation, which tests whether any D
is a divisor, yields a quadratic algorithm. A better approach,
inspired by Eratosthene’s sieve, is the following: at step j, we know
C[j], and we subtract it from all multiples of j. The running time of
the subroutine becomes O(N log N).
Relevant Code
#include <cassert>
#include <algorithm>
#include <fstream>
#include <iostream>
#include <vector>
using namespace std;
const int kMaxN = 2e5;
typedef int int32;
typedef long long int64_x;
// #define int __int128_t
// #define int64 __int128_t
typedef long long int64;
int64 count_less(int a, int n) {
vector<int> counter(n + 1, 0);
for (int i = 2; i <= n; i += 1) {
counter[i] = min(1LL * (i - 1), 1LL * i * a / n);
}
int64 result = 0;
for (int i = 2; i <= n; i += 1) {
for (int j = 2 * i; j <= n; j += i) {
counter[j] -= counter[i];
}
result += counter[i];
}
return result;
}
int32 main() {
// ifstream cin("farey.in");
// ofstream cout("farey.out");
int64_x n, k; cin >> n >> k;
assert(1 <= n);
assert(n <= kMaxN);
assert(1 <= k);
assert(k <= count_less(n, n));
int up = 0;
for (int p = 29; p >= 0; p -= 1) {
if ((1 << p) + up > n)
continue;
if (count_less((1 << p) + up, n) < k) {
up += (1 << p);
}
}
k -= count_less(up, n);
vector<pair<int, int>> elements;
for (int i = 1; i <= n; i += 1) {
int b = i;
// find a such that up/n < a / b and a / b <= (up+1) / n
int a = 1LL * (up + 1) * b / n;
if (1LL * up * b < 1LL * a * n) {
} else {
continue;
}
if (1LL * a * n <= 1LL * (up + 1) * b) {
} else {
continue;
}
if (__gcd(a, b) != 1) {
continue;
}
elements.push_back({a, b});
}
sort(elements.begin(), elements.end(),
[](const pair<int, int>& lhs, const pair<int, int>& rhs) -> bool {
return 1LL * lhs.first * rhs.second < 1LL * rhs.first * lhs.second;
});
cout << (int64_x)elements[k - 1].first << ' ' << (int64_x)elements[k - 1].second << '\n';
return 0;
}
Basic Methodology
The above editorial explanation results in the following simplified version. Let me start with an example.
Let's say, we want to find 7th element of Farey Sequence with N = 5.
We start with writing a subroutine, as said in the explanation, that gives us the "k" value (how many Farey Sequence reduced fractions there exist before a given fraction - the given number may or may not be reduced)
So, take your F5 sequence:
k = 0, 0/1
k = 1, 1/5
k = 2, 1/4
k = 3, 1/3
k = 4, 2/5
k = 5, 1/2
k = 6, 3/5
k = 7, 2/3
k = 8, 3/4
k = 9, 4/5
k = 10, 1/1
If we can find a function that finds the count of the previous reduced fractions in Farey Sequence, we can do the following:
int64 k_count_2 = count_less(2, 5); // result = 4
int64 k_count_3 = count_less(3, 5); // result = 6
int64 k_count_4 = count_less(4, 5); // result = 9
This function is written in the accepted solution. It uses the exact methodology explained in the last paragraph of the editorial.
As you can see, the count_less() function generates the same k values as in our hand written list.
We know the values of the reduced fractions for k = 4, 6, 9 using that function. What about k = 7? As explained in the editorial, we will list all the reduced fractions in range X/N and (X+1)/N, here X = 3 and N = 5.
Using the function in the accepted solution (its near bottom), we list and sort the reduced fractions.
After that we will rearrange our k values, as in to fit in our new array as such:
k = -, 0/1
k = -, 1/5
k = -, 1/4
k = -, 1/3
k = -, 2/5
k = -, 1/2
k = -, 3/5 <-|
k = 0, 2/3 | We list and sort the possible reduced fractions
k = 1, 3/4 | in between these numbers
k = -, 4/5 <-|
k = -, 1/1
(That's why there is this piece of code: k -= count_less(up, n);, it basically remaps the k values)
(And we also subtract one more during indexing, i.e.: cout << (int64_x)elements[k - 1].first << ' ' << (int64_x)elements[k - 1].second << '\n';. This is just to basically call the right position in the generated array.)
So, for our new re-mapped k values, for N = 5 and k = 7 (original k), our result is 2/3.
(We select the value k = 0, in our new map)
If you compile and run the accepted solution, it will give you this:
Input: 5 7 (Enter)
Output: 2 3
I believe this is the basic point of the editorial and accepted solution.
Title says it all.
I need to split n as sum of k parts where each part ki should be in the range of
1 <= ki <= ri for given array r.
for example -
n = 4, k = 3 and r = [2, 2, 1]
ans = 2
#[2, 1, 1], [1, 2, 1]
Order matters. (2, 1, 1) and (1, 2, 1) are different.
I taught of solving it using stars and bars method, but be because of upper bound ri i dont know to to approach it.
i implemented a direct recursion function and it works fine for small values only.
Constraints of original problem are
1 <= n <= 107
1 <= k <= 105
1 <= ri <= 51
All calculations will be done under prime Modulo.
i found a similar problem here but i don't know how to implement in program. HERE
My brute-force recursive function -
#define MAX 1000
const int md = 1e9 + 7;
vector <int> k;
vector <map<int, int>> mapper;
vector <int> hold;
int solve(int sum, int cur){
if(cur == (k.size() - 1) && sum >= 1 && sum <= k[cur]) return 1;
if(cur == (k.size() - 1) && (sum < 1 || sum > k[cur])) return 0;
if(mapper[cur].find(sum) != mapper[cur].end())
return mapper[cur][sum];
int ans = 0;
int start = 1;
for(int i=start; i<=k[cur]; ++i){
int remain = sum - i;
int seg = (k.size() - cur) - 1;
if(remain < seg) break;
int res = solve(sum - i, cur + 1);
ans = (1LL * ans + res) % md;
}
mapper[cur][sum] = ans;
return ans;
}
int main(){
for(int i=0; i<MAX; ++i) k.push_back(51); // restriction for each part default 51
mapper.resize(MAX);
cout << solve(MAX + MAX, 0) << endl;
}
Instead of using a map for storing result of computation i used a two dimensional array and it gave very good performance boost but i cannot use it because of large n and k values.
How could i improve my recursive function or what are other ways of solving this problem.
That's interesting problem.
First lets say r_i = r_i - 1, n = n - k, numbers in [0, r_i] just for convenience. Now it's possible to add some fictitious numbers to make m the power of 2 without changing answer.
Now let's represent each interval of [0, r_i] as polynomial 1 * x ^ 0 + 1 * x ^ 1 + ... + 1 * x & r_i. Now if we multiply all these polynomials, coefficient at x ^ n will be answer.
Here is structure called Number Theoretic Transform (NTT) which allows to multiply two polynomials modulo p in O(size * log(size)).
If you will just multiply it using NTT, code will work in something like O(n * k * log (k * max(r))). It's very slow.
But now our fictive numbers help. Let's use divide and conquer technics. We'll make O(log m) steps, on each step multiply 2 * i-th and 2 * i + 1-th polynomials. In the next step we'll multiply resulting polynomials of this step.
Each step works in O(k * log(k)) and there is O(log(k)) steps, so algorhitm works in O(k * log^2 (k)). It's fast asymptotically, but I'm not sure if it fits TL for this problem. I think it will work about 20 seconds on max test.
In an attempt to solve the 3rd problem on project Euler (https://projecteuler.net/problem=3), I decided to implement Pollard's Rho algorithm (at least part of it, I'm planning on including the cycling later). The odd thing is that it works for numbers such as: 82123(factor = 41) and 16843009(factor 257). However when I try the project Euler number: 600851475143, I end up getting 71 when the largest prime factor is 6857. Here's my implementation(sorry for wall of code and lack of type casting):
#include <iostream>
#include <math.h>
#include <vector>
using namespace std;
long long int gcd(long long int a,long long int b);
long long int f(long long int x);
int main()
{
long long int i, x, y, N, factor, iterations = 0, counter = 0;
vector<long long int>factors;
factor = 1;
x = 631;
N = 600851475143;
factors.push_back(x);
while (factor == 1)
{
y = f(x);
y = y % N;
factors.push_back(y);
cout << "\niteration" << iterations << ":\t";
i = 0;
while (factor == 1 && (i < factors.size() - 1))
{
factor = gcd(abs(factors.back() - factors[i]), N);
cout << factor << " ";
i++;
}
x = y;
//factor = 2;
iterations++;
}
system("PAUSE");
return 0;
}
long long int gcd(long long int a, long long int b)
{
long long int remainder;
do
{
remainder = a % b;
a = b;
b = remainder;
} while (remainder != 0);
return a;
}
long long int f(long long int x)
{
//x = x*x * 1024 + 32767;
x = x*x + 1;
return x;
}
Pollard's rho algorithm guarantees nothing. It doesn't guarantee to find the largest factor. It doesn't guarantee that any factor it finds is prime. It doesn't even guarantee to find a factor at all. The rho algorithm is probabilistic; it will probably find a factor, but not necessarily. Since your function returns a factor, it works.
That said, your implementation isn't very good. It's not necessary to store all previous values of the function, and compute the gcd to each every time through the loop. Here is pseudocode for a better version of the function:
function rho(n)
for c from 1 to infinity
h, t := 1, 1
repeat
h := (h*h+c) % n # the hare runs ...
h := (h*h+c) % n # ... twice as fast
t := (t*t+c) % n # as the tortoise
g := gcd(t-h, n)
while g == 1
if g < n then return g
This function returns a single factor of n, which may be either prime or composite. It stores only two values of the random sequence, and stops when it finds a cycle (when g == n), restarting with a different random sequence (by incrementing c). Otherwise it keeps going until it finds a factor, which shouldn't take too long as long as you limit the input to 64-bit integers. Find more factors by applying rho to the remaining cofactor, or if the factor that is found is composite, stopping when all the prime factors have been found.
By the way, you don't need Pollard's rho algorithm to solve Project Euler #3; simple trial division is sufficient. This algorithm finds all the prime factors of a number, from which you can extract the largest:
function factors(n)
f := 2
while f * f <= n
while n % f == 0
print f
n := n / f
f := f + 1
if n > 1 then print n
I need to calculate combinations for a number.
What is the fastest way to calculate nCp where n>>p?
I need a fast way to generate binomial coefficients for an polynomial equation and I need to get the coefficient of all the terms and store it in an array.
(a+b)^n = a^n + nC1 a^(n-1) * b + nC2 a^(n-2) * ............
+nC(n-1) a * b^(n-1) + b^n
What is the most efficient way to calculate nCp ??
You cau use dynamic programming in order to generate binomial coefficients
You can create an array and than use O(N^2) loop to fill it
C[n, k] = C[n-1, k-1] + C[n-1, k];
where
C[1, 1] = C[n, n] = 1
After that in your program you can get the C(n, k) value just looking at your 2D array at [n, k] indices
UPDATE smth like that
for (int k = 1; k <= K; k++) C[0][k] = 0;
for (int n = 0; n <= N; n++) C[n][0] = 1;
for (int n = 1; n <= N; n++)
for (int k = 1; k <= K; k++)
C[n][k] = C[n-1][k-1] + C[n-1][k];
where the N, K - maximum values of your n, k
If you need to compute them for all n, Ribtoks's answer is probably the best.
For a single n, you're better off doing like this:
C[0] = 1
for (int k = 0; k < n; ++ k)
C[k+1] = (C[k] * (n-k)) / (k+1)
The division is exact, if done after the multiplication.
And beware of overflowing with C[k] * (n-k) : use large enough integers.
If you want complete expansions for large values of n, FFT convolution might be the fastest way. In the case of a binomial expansion with equal coefficients (e.g. a series of fair coin tosses) and an even order (e.g. number of tosses) you can exploit symmetries thus:
Theory
Represent the results of two coin tosses (e.g. half the difference between the total number of heads and tails) with the expression A + A*cos(Pi*n/N). N is the number of samples in your buffer - a binomial expansion of even order O will have O+1 coefficients and require a buffer of N >= O/2 + 1 samples - n is the sample number being generated, and A is a scale factor that will usually be either 2 (for generating binomial coefficients) or 0.5 (for generating a binomial probability distribution).
Notice that, in frequency, this expression resembles the binomial distribution of those two coin tosses - there are three symmetrical spikes at positions corresponding to the number (heads-tails)/2. Since modelling the overall probability distribution of independent events requires convolving their distributions, we want to convolve our expression in the frequency domain, which is equivalent to multiplication in the time domain.
In other words, by raising our cosine expression for the result of two tosses to a power (e.g. to simulate 500 tosses, raise it to the power of 250 since it already represents a pair), we can arrange for the binomial distribution for a large number to appear in the frequency domain. Since this is all real and even, we can substitute the DCT-I for the DFT to improve efficiency.
Algorithm
decide on a buffer size, N, that is at least O/2 + 1 and can be conveniently DCTed
initialise it with the expression pow(A + A*cos(Pi*n/N),O/2)
apply the forward DCT-I
read out the coefficients from the buffer - the first number is the central peak where heads=tails, and subsequent entries correspond to symmetrical pairs successively further from the centre
Accuracy
There's a limit to how high O can be before accumulated floating-point rounding errors rob you of accurate integer values for the coefficients, but I'd guess the number is pretty high. Double-precision floating-point can represent 53-bit integers with complete accuracy, and I'm going to ignore the rounding loss involved in the use of pow() because the generating expression will take place in FP registers, giving us an extra 11 bits of mantissa to absorb the rounding error on Intel platforms. So assuming we use a 1024-point DCT-I implemented via the FFT, that means losing 10 bits' accuracy to rounding error during the transform and not much else, leaving us with ~43 bits of clean representation. I don't know what order of binomial expansion generates coefficients of that size, but I dare say it's big enough for your needs.
Asymmetrical expansions
If you want the asymmetrical expansions for unequal coefficients of a and b, you'll need to use a two-sided (complex) DFT and a complex pow() function. Generate the expression A*A*e^(-Pi*i*n/N) + A*B + B*B*e^(+Pi*i*n/N) [using the complex pow() function to raise it to the power of half the expansion order] and DFT it. What you have in the buffer is, again, the central point (but not the maximum if A and B are very different) at offset zero, and it is followed by the upper half of the distribution. The upper half of the buffer will contain the lower half of the distribution, corresponding to heads-minus-tails values that are negative.
Notice that the source data is Hermitian symmetrical (the second half of the input buffer is the complex conjugate of the first), so this algorithm is not optimal and can be performed using a complex-to-complex FFT of half the required size for optimum efficiency.
Needless to say, all the complex exponentiation will chew more CPU time and hurt accuracy compared to the purely real algorithm for symmetrical distributions above.
This is my version:
def binomial(n, k):
if k == 0:
return 1
elif 2*k > n:
return binomial(n,n-k)
else:
e = n-k+1
for i in range(2,k+1):
e *= (n-k+i)
e /= i
return e
I recently wrote a piece of code that needed to call for a binary coefficient about 10 million times. So I did a combination lookup-table/calculation approach that's still not too wasteful of memory. You might find it useful (and my code is in the public domain). The code is at
http://www.etceterology.com/fast-binomial-coefficients
It's been suggested that I inline the code here. A big honking lookup table seems like a waste, so here's the final function, and a Python script that generates the table:
extern long long bctable[]; /* See below */
long long binomial(int n, int k) {
int i;
long long b;
assert(n >= 0 && k >= 0);
if (0 == k || n == k) return 1LL;
if (k > n) return 0LL;
if (k > (n - k)) k = n - k;
if (1 == k) return (long long)n;
if (n <= 54 && k <= 54) {
return bctable[(((n - 3) * (n - 3)) >> 2) + (k - 2)];
}
/* Last resort: actually calculate */
b = 1LL;
for (i = 1; i <= k; ++i) {
b *= (n - (k - i));
if (b < 0) return -1LL; /* Overflow */
b /= i;
}
return b;
}
#!/usr/bin/env python3
import sys
class App(object):
def __init__(self, max):
self.table = [[0 for k in range(max + 1)] for n in range(max + 1)]
self.max = max
def build(self):
for n in range(self.max + 1):
for k in range(self.max + 1):
if k == 0: b = 1
elif k > n: b = 0
elif k == n: b = 1
elif k == 1: b = n
elif k > n-k: b = self.table[n][n-k]
else:
b = self.table[n-1][k] + self.table[n-1][k-1]
self.table[n][k] = b
def output(self, val):
if val > 2**63: val = -1
text = " {0}LL,".format(val)
if self.column + len(text) > 76:
print("\n ", end = "")
self.column = 3
print(text, end = "")
self.column += len(text)
def dump(self):
count = 0
print("long long bctable[] = {", end="");
self.column = 999
for n in range(self.max + 1):
for k in range(self.max + 1):
if n < 4 or k < 2 or k > n-k:
continue
self.output(self.table[n][k])
count += 1
print("\n}}; /* {0} Entries */".format(count));
def run(self):
self.build()
self.dump()
return 0
def main(args):
return App(54).run()
if __name__ == "__main__":
sys.exit(main(sys.argv))
If you really only need the case where n is much larger than p, one way to go would be to use the Stirling's formula for the factorials. (if n>>1 and p is order one, Stirling approximate n! and (n-p)!, keep p! as it is etc.)
The fastest reasonable approximation in my own benchmarking is the approximation used by the Apache Commons Maths library: http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/special/Gamma.html#logGamma(double)
My colleagues and I tried to see if we could beat it, while using exact calculations rather than approximates. All approaches failed miserably (many orders slower) except one, which was 2-3 times slower. The best performing approach uses https://math.stackexchange.com/a/202559/123948, here is the code (in Scala):
var i: Int = 0
var binCoeff: Double = 1
while (i < k) {
binCoeff *= (n - i) / (k - i).toDouble
i += 1
}
binCoeff
The really bad approaches where various attempts at implementing Pascal's Triangle using tail recursion.
nCp = n! / ( p! (n-p)! ) =
( n * (n-1) * (n-2) * ... * (n - p) * (n - p - 1) * ... * 1 ) /
( p * (p-1) * ... * 1 * (n - p) * (n - p - 1) * ... * 1 )
If we prune the same terms of the numerator and the denominator, we are left with minimal multiplication required. We can write a function in C to perform 2p multiplications and 1 division to get nCp:
int binom ( int p, int n ) {
if ( p == 0 ) return 1;
int num = n;
int den = p;
while ( p > 1 ) {
p--;
num *= n - p;
den *= p;
}
return num / den;
}
I was looking for the same thing and couldn't find it, so wrote one myself that seems optimal for any Binomial Coeffcient for which the endresult fits into a Long.
// Calculate Binomial Coefficient
// Jeroen B.P. Vuurens
public static long binomialCoefficient(int n, int k) {
// take the lowest possible k to reduce computing using: n over k = n over (n-k)
k = java.lang.Math.min( k, n - k );
// holds the high number: fi. (1000 over 990) holds 991..1000
long highnumber[] = new long[k];
for (int i = 0; i < k; i++)
highnumber[i] = n - i; // the high number first order is important
// holds the dividers: fi. (1000 over 990) holds 2..10
int dividers[] = new int[k - 1];
for (int i = 0; i < k - 1; i++)
dividers[i] = k - i;
// for every dividers there is always exists a highnumber that can be divided by
// this, the number of highnumbers being a sequence that equals the number of
// dividers. Thus, the only trick needed is to divide in reverse order, so
// divide the highest divider first trying it on the highest highnumber first.
// That way you do not need to do any tricks with primes.
for (int divider: dividers) {
boolean eliminated = false;
for (int i = 0; i < k; i++) {
if (highnumber[i] % divider == 0) {
highnumber[i] /= divider;
eliminated = true;
break;
}
}
if(!eliminated) throw new Error(n+","+k+" divider="+divider);
}
// multiply remainder of highnumbers
long result = 1;
for (long high : highnumber)
result *= high;
return result;
}
If I understand the notation in the question, you don't just want nCp, you actually want all of nC1, nC2, ... nC(n-1). If this is correct, we can leverage the following relationship to make this fairly trivial:
for all k>0: nCk = prod_{from i=1..k}( (n-i+1)/i )
i.e. for all k>0: nCk = nC(k-1) * (n-k+1) / k
Here's a python snippet implementing this approach:
def binomial_coef_seq(n, k):
"""Returns a list of all binomial terms from choose(n,0) up to choose(n,k)"""
b = [1]
for i in range(1,k+1):
b.append(b[-1] * (n-i+1)/i)
return b
If you need all coefficients up to some k > ceiling(n/2), you can use symmetry to reduce the number of operations you need to perform by stopping at the coefficient for ceiling(n/2) and then just backfilling as far as you need.
import numpy as np
def binomial_coef_seq2(n, k):
"""Returns a list of all binomial terms from choose(n,0) up to choose(n,k)"""
k2 = int(np.ceiling(n/2))
use_symmetry = k > k2
if use_symmetry:
k = k2
b = [1]
for i in range(1, k+1):
b.append(b[-1] * (n-i+1)/i)
if use_symmetry:
v = k2 - (n-k)
b2 = b[-v:]
b.extend(b2)
return b
Time Complexity : O(denominator)
Space Complexity : O(1)
public class binomialCoeff {
static double binomialcoeff(int numerator, int denominator)
{
double res = 1;
//invalid numbers
if (denominator>numerator || denominator<0 || numerator<0) {
res = -1;
return res;}
//default values
if(denominator==numerator || denominator==0 || numerator==0)
return res;
// Since C(n, k) = C(n, n-k)
if ( denominator > (numerator - denominator) )
denominator = numerator - denominator;
// Calculate value of [n * (n-1) *---* (n-k+1)] / [k * (k-1) *----* 1]
while (denominator>=1)
{
res *= numerator;
res = res / denominator;
denominator--;
numerator--;
}
return res;
}
/* Driver program to test above function*/
public static void main(String[] args)
{
int numerator = 120;
int denominator = 20;
System.out.println("Value of C("+ numerator + ", " + denominator+ ") "
+ "is" + " "+ binomialcoeff(numerator, denominator));
}
}
I saw this question, and pop up this idea.
There exists a constant time (pretty fast) method for integers of limited size (e.g. 32-bit integers).
Note that for an integer N that is a power of 3 the following is true:
For any M <= N that is a power of 3, M divides N.
For any M <= N that is not a power 3, M does not divide N.
The biggest power of 3 that fits into 32 bits is 3486784401 (3^20). This gives the following code:
bool isPower3(std::uint32_t value) {
return value != 0 && 3486784401u % value == 0;
}
Similarly for signed 32 bits it is 1162261467 (3^19):
bool isPower3(std::int32_t value) {
return value > 0 && 1162261467 % value == 0;
}
In general the magic number is:
== pow(3, floor(log(MAX) / log(3)))
Careful with floating point rounding errors, use a math calculator like Wolfram Alpha to calculate the constant. For example for 2^63-1 (signed int64) both C++ and Java give 4052555153018976256, but the correct value is 4052555153018976267.
while (n % 3 == 0) {
n /= 3;
}
return n == 1;
Note that 1 is the zeroth power of three.
Edit: You also need to check for zero before the loop, as the loop will not terminate for n = 0 (thanks to Bruno Rothgiesser).
I find myself slightly thinking that if by 'integer' you mean 'signed 32-bit integer', then (pseudocode)
return (n == 1)
or (n == 3)
or (n == 9)
...
or (n == 1162261467)
has a certain beautiful simplicity to it (the last number is 3^19, so there aren't an absurd number of cases). Even for an unsigned 64-bit integer there still be only 41 cases (thanks #Alexandru for pointing out my brain-slip). And of course would be impossible for arbitrary-precision arithmetic...
I'm surprised at this. Everyone seems to have missed the fastest algorithm of all.
The following algorithm is faster on average - and dramatically faster in some cases - than a simple while(n%3==0) n/=3; loop:
bool IsPowerOfThree(uint n)
{
// Optimizing lines to handle the most common cases extremely quickly
if(n%3 != 0) return n==1;
if(n%9 != 0) return n==3;
// General algorithm - works for any uint
uint r;
n = Math.DivRem(n, 59049, out r); if(n!=0 && r!=0) return false;
n = Math.DivRem(n+r, 243, out r); if(n!=0 && r!=0) return false;
n = Math.DivRem(n+r, 27, out r); if(n!=0 && r!=0) return false;
n += r;
return n==1 || n==3 || n==9;
}
The numeric constants in the code are 3^10, 3^5, and 3^3.
Performance calculations
In modern CPUs, DivRem is a often single instruction that takes a one cycle. On others it expands to a div followed by a mul and an add, which would takes more like three cycles altogether. Each step of the general algorithm looks long but it actually consists only of: DivRem, cmp, cmove, cmp, cand, cjmp, add. There is a lot of parallelism available, so on a typical two-way superscalar processor each step will likely execute in about 4 clock cycles, giving a guaranteed worst-case execution time of about 25 clock cycles.
If input values are evenly distributed over the range of UInt32, here are the probabilities associated with this algorithm:
Return in or before the first optimizing line: 66% of the time
Return in or before the second optimizing line: 89% of the time
Return in or before the first general algorithm step: 99.998% of the time
Return in or before the second general algorithm step: 99.99998% of the time
Return in or before the third general algorithm step: 99.999997% of the time
This algorithm outperforms the simple while(n%3==0) n/=3 loop, which has the following probabilities:
Return in the first iteration: 66% of the time
Return in the first two iterations: 89% of the time
Return in the first three iterations: 97% of the time
Return in the first four iterations: 98.8% of the time
Return in the first five iterations: 99.6% of the time ... and so on to ...
Return in the first twelve iterations: 99.9998% of the time ... and beyond ...
What is perhaps even more important, this algorithm handles midsize and large powers of three (and multiples thereof) much more efficiently: In the worst case the simple algorithm will consume over 100 CPU cycles because it will loop 20 times (41 times for 64 bits). The algorithm I present here will never take more than about 25 cycles.
Extending to 64 bits
Extending the above algorithm to 64 bits is trivial - just add one more step. Here is a 64 bit version of the above algorithm optimized for processors without efficient 64 bit division:
bool IsPowerOfThree(ulong nL)
{
// General algorithm only
ulong rL;
nL = Math.DivRem(nL, 3486784401, out rL); if(nL!=0 && rL!=0) return false;
nL = Math.DivRem(nL+rL, 59049, out rL); if(nL!=0 && rL!=0) return false;
uint n = (uint)nL + (uint)rL;
n = Math.DivRem(n, 243, out r); if(n!=0 && r!=0) return false;
n = Math.DivRem(n+r, 27, out r); if(n!=0 && r!=0) return false;
n += r;
return n==1 || n==3 || n==9;
}
The new constant is 3^20. The optimization lines are omitted from the top of the method because under our assumption that 64 bit division is slow, they would actually slow things down.
Why this technique works
Say I want to know if "100000000000000000" is a power of 10. I might follow these steps:
I divide by 10^10 and get a quotient of 10000000 and a remainder of 0. These add to 10000000.
I divide by 10^5 and get a quotient of 100 and a remainder of 0. These add to 100.
I divide by 10^3 and get a quotient of 0 and a remainderof 100. These add to 100.
I divide by 10^2 and get a quotient of 1 and a remainder of 0. These add to 1.
Because I started with a power of 10, every time I divided by a power of 10 I ended up with either a zero quotient or a zero remainder. Had I started out with anything except a power of 10 I would have sooner or later ended up with a nonzero quotient or remainder.
In this example I selected exponents of 10, 5, and 3 to match the code provided previously, and added 2 just for the heck of it. Other exponents would also work: There is a simple algorithm for selecting the ideal exponents given your maximum input value and the maximum power of 10 allowed in the output, but this margin does not have enough room to contain it.
NOTE: You may have been thinking in base ten throughout this explanation, but the entire explanation above can be read and understood identically if you're thinking in in base three, except the exponents would have been expressed differently (instead of "10", "5", "3" and "2" I would have to say "101", "12", "10" and "2").
This is a summary of all good answers below this questions, and the performance figures can be found from the LeetCode article.
1. Loop Iteration
Time complexity O(log(n)), space complexity O(1)
public boolean isPowerOfThree(int n) {
if (n < 1) {
return false;
}
while (n % 3 == 0) {
n /= 3;
}
return n == 1;
}
2. Base Conversion
Convert the integer to a base 3 number, and check if it is written as a leading 1 followed by all 0. It is inspired by the solution to check if a number is power of 2 by doing n & (n - 1) == 0
Time complexity: O(log(n)) depending on language and compiler, space complexity: O(log(n))
public boolean isPowerOfThree(int n) {
return Integer.toString(n, 3).matches("^10*$");
}
3. Mathematics
If n = 3^i, then i = log(n) / log(3), and thus comes to the solution
Time complexity: depending on language and compiler, space complexity: O(1)
public boolean isPowerOfThree(int n) {
return (Math.log(n) / Math.log(3) + epsilon) % 1 <= 2 * epsilon;
}
4. Integer Limitations
Because 3^19 = 1162261467 is the largest power of 3 number fits in a 32 bit integer, thus we can do
Time complexity: O(1), space complexity: O(1)
public boolean isPowerOfThree(int n) {
return n > 0 && 1162261467 % n == 0;
}
5. Integer Limitations with Set
The idea is similar to #4 but use a set to store all possible power of 3 numbers (from 3^0 to 3^19). It makes code more readable.
6. Recursive (C++11)
This solution is specific to C++11, using template meta programming so that complier will replace the call isPowerOf3<Your Input>::cValue with calculated result.
Time complexity: O(1), space complexity: O(1)
template<int N>
struct isPowerOf3 {
static const bool cValue = (N % 3 == 0) && isPowerOf3<N / 3>::cValue;
};
template<>
struct isPowerOf3<0> {
static const bool cValue = false;
};
template<>
struct isPowerOf3<1> {
static const bool cValue = true;
};
int main() {
cout<<isPowerOf3<1162261467>::cValue;
return 0;
}
if (log n) / (log 3) is integral then n is a power of 3.
Recursively divide by 3, check that the remainder is zero and re-apply to the quotient.
Note that 1 is a valid answer as 3 to the zero power is 1 is an edge case to beware.
Very interesting question, I like the answer from starblue,
and this is a variation of his algorithm which will converge little bit faster to the solution:
private bool IsPow3(int n)
{
if (n == 0) return false;
while (n % 9 == 0)
{
n /= 9;
}
return (n == 1 || n == 3);
}
Between powers of two there is at most one power of three.
So the following is a fast test:
Find the binary logarithm of n by finding the position of the leading 1 bit in the number. This is very fast, as modern processors have a special instruction for that. (Otherwise you can do it by bit twiddling, see Bit Twiddling Hacks).
Look up the potential power of three in a table indexed by this position and compare to n (if there is no power of three you can store any number with a different binary logarithm).
If they are equal return yes, otherwise no.
The runtime depends mostly on the time needed for accessing the table entry. If we are using machine integers the table is small, and probably in cache (we are using it many millions of times, otherwise this level of optimization wouldn't make sense).
Here is a nice and fast implementation of Ray Burns' method in C:
bool is_power_of_3(unsigned x) {
if (x > 0x0000ffff)
x *= 0xb0cd1d99; // multiplicative inverse of 59049
if (x > 0x000000ff)
x *= 0xd2b3183b; // multiplicative inverse of 243
return x <= 243 && ((x * 0x71c5) & 0x5145) == 0x5145;
}
It uses the multiplicative inverse trick for to first divide by 3^10 and then by 3^5. Finally, it needs to check whether the result is 1, 3, 9, 27, 81, or 243, which is done by some simple hashing that I found by trial-and-error.
On my CPU (Intel Sandy Bridge), it is quite fast, but not as fast as the method of starblue that uses the binary logarithm (which is implemented in hardware on that CPU). But on a CPU without such an instruction, or when lookup tables are undesirable, it might be an alternative.
How large is your input? With O(log(N)) memory you can do faster, O(log(log(N)). Precompute the powers of 3 and then do a binary search on the precomputed values.
Simple and constant-time solution:
return n == power(3, round(log(n) / log(3)))
For really large numbers n, you can use the following math trick to speed up the operation of
n % 3 == 0
which is really slow and most likely the choke point of any algorithm that relies on repeated checking of remainders. You have to understand modular arithmetic to follow what I am doing, which is part of elementary number theory.
Let x = Σ k a k 2 k be the number of interest. We can let the upper bound of the sum be ∞ with the understanding that a k = 0 for some k > M. Then
0 ≡ x ≡ Σ k a k 2 k ≡ Σ k a 2k 2 2k + a 2k+1 2 2k+1 ≡ Σ k 2 2k ( a 2k + a 2k+1 2) ≡ Σ k a 2k + a 2k+1 2 (mod 3)
since 22k ≡ 4 k ≡ 1k ≡ 1 (mod 3).
Given a binary representation of a number x with 2n+1 bits as
x0 x1 x2 ... x2n+1
where xk ∈{0,1} you can group odd even pairs
(x0 x1) (x2 x3) ... (x2n x2n+1).
Let q denote the number of pairings of the form (1 0) and let r denote the number of pairings of the form (0 1). Then it follows from the equation above that 3 | x if and only if 3 | (q + 2r). Furthermore, you can show that 3|(q + 2r) if and only if q and r have the same remainder when divided by 3.
So an algorithm for determining whether a number is divisible by 3 could be done as follows
q = 0, r = 0
for i in {0,1, .., n}
pair <- (x_{2i} x_{2i+1})
if pair == (1 0)
switch(q)
case 0:
q = 1;
break;
case 1:
q = 2;
break;
case 2:
q = 0;
break;
else if pair == (0 1)
switch(r)
case 0:
r = 1;
break;
case 1:
r = 2;
break;
case 2:
r = 0;
return q == r
This algorithm is more efficient than the use of %.
--- Edit many years later ----
I took a few minutes to implement a rudimentary version of this in python that checks its true for all numbers up to 10^4. I include it below for reference. Obviously, to make use of this one would implement this as close to hardware as possible. This scanning technique can be extended to any number that one wants to by altering the derivation. I also conjecture the 'scanning' portion of the algorithm can be reformulated in a recursive O(log n) type formulation similar to a FFT, but I'd have to think on it.
#!/usr/bin/python
def bits2num(bits):
num = 0
for i,b in enumerate(bits):
num += int(b) << i
return num
def num2bits(num):
base = 0
bits = list()
while True:
op = 1 << base
if op > num:
break
bits.append(op&num !=0)
base += 1
return "".join(map(str,map(int,bits)))[::-1]
def div3(bits):
n = len(bits)
if n % 2 != 0:
bits = bits + '0'
n = len(bits)
assert n % 2 == 0
q = 0
r = 0
for i in range(n/2):
pair = bits[2*i:2*i+2]
if pair == '10':
if q == 0:
q = 1
elif q == 1:
q = 2
elif q == 2:
q = 0
elif pair == '01':
if r == 0:
r = 1
elif r == 1:
r = 2
elif r == 2:
r = 0
else:
pass
return q == r
for i in range(10000):
truth = (i % 3) == 0
bits = num2bits(i)
check = div3(bits)
assert truth == check
You can do better than repeated division, which takes O(lg(X) * |division|) time. Essentially you do a binary search on powers of 3. Really we will be doing a binary search on N, where 3^N = input value). Setting the Pth binary digit of N corresponds to multiplying by 3^(2^P), and values of the form 3^(2^P) can be computed by repeated squaring.
Algorithm
Let the input value be X.
Generate a list L of repeated squared values which ends once you pass X.
Let your candidate value be T, initialized to 1.
For each E in reversed L, if T*E <= X then let T *= E.
Return T == X.
Complexity:
O(lg(lg(X)) * |multiplication|)
- Generating and iterating over L takes lg(lg(X)) iterations, and multiplication is the most expensive operation in an iteration.
The fastest solution is either testing if n > 0 && 3**19 % n == 0 as given in another answer or perfect hashing (below). First I'm giving two multiplication-based solutions.
Multiplication
I wonder why everybody missed that multiplication is much faster than division:
for (int i=0, pow=1; i<=19, pow*=3; ++i) {
if (pow >= n) {
return pow == n;
}
}
return false;
Just try all powers, stop when it grew too big. Avoid overflow as 3**19 = 0x4546B3DB is the biggest power fitting in signed 32-bit int.
Multiplication with binary search
Binary search could look like
int pow = 1;
int next = pow * 6561; // 3**8
if (n >= next) pow = next;
next = pow * 81; // 3**4
if (n >= next) pow = next;
next = pow * 81; // 3**4; REPEATED
if (n >= next) pow = next;
next = pow * 9; // 3**2
if (n >= next) pow = next;
next = pow * 3; // 3**1
if (n >= next) pow = next;
return pow == next;
One step is repeated, so that the maximum exponent 19 = 8+4+4+2+1 can exactly be reached.
Perfect hashing
There are 20 powers of three fitting into a signed 32-bit int, so we take a table of 32 elements. With some experimentation, I found the perfect hash function
def hash(x):
return (x ^ (x>>1) ^ (x>>2)) & 31;
mapping each power to a distinct index between 0 and 31. The remaining stuff is trivial:
// Create a table and fill it with some power of three.
table = [1 for i in range(32)]
// Fill the buckets.
for n in range(20): table[hash(3**n)] = 3**n;
Now we have
table = [
1162261467, 1, 3, 729, 14348907, 1, 1, 1,
1, 1, 19683, 1, 2187, 81, 1594323, 9,
27, 43046721, 129140163, 1, 1, 531441, 243, 59049,
177147, 6561, 1, 4782969, 1, 1, 1, 387420489]
and can test very fast via
def isPowerOfThree(x):
return table[hash(x)] == x
Your question is fairly easy to answer by defining a simple function to run the check for you. The example implementation shown below is written in Python but should not be difficult to rewrite in other languages if needed. Unlike the last version of this answer, the code shown below is far more reliable.
Python 3.6.0 (v3.6.0:41df79263a11, Dec 23 2016, 08:06:12) [MSC v.1900 64 bit (AMD64)] on win32
Type "copyright", "credits" or "license()" for more information.
>>> import math
>>> def power_of(number, base):
return number == base ** round(math.log(number, base))
>>> base = 3
>>> for power in range(21):
number = base ** power
print(f'{number} is '
f'{"" if power_of(number, base) else "not "}'
f'a power of {base}.')
number += 1
print(f'{number} is '
f'{"" if power_of(number, base) else "not "}'
f'a power of {base}.')
print()
1 is a power of 3.
2 is not a power of 3.
3 is a power of 3.
4 is not a power of 3.
9 is a power of 3.
10 is not a power of 3.
27 is a power of 3.
28 is not a power of 3.
81 is a power of 3.
82 is not a power of 3.
243 is a power of 3.
244 is not a power of 3.
729 is a power of 3.
730 is not a power of 3.
2187 is a power of 3.
2188 is not a power of 3.
6561 is a power of 3.
6562 is not a power of 3.
19683 is a power of 3.
19684 is not a power of 3.
59049 is a power of 3.
59050 is not a power of 3.
177147 is a power of 3.
177148 is not a power of 3.
531441 is a power of 3.
531442 is not a power of 3.
1594323 is a power of 3.
1594324 is not a power of 3.
4782969 is a power of 3.
4782970 is not a power of 3.
14348907 is a power of 3.
14348908 is not a power of 3.
43046721 is a power of 3.
43046722 is not a power of 3.
129140163 is a power of 3.
129140164 is not a power of 3.
387420489 is a power of 3.
387420490 is not a power of 3.
1162261467 is a power of 3.
1162261468 is not a power of 3.
3486784401 is a power of 3.
3486784402 is not a power of 3.
>>>
NOTE: The last revision has caused this answer to become nearly the same as TMS' answer.
Set based solution...
DECLARE #LastExponent smallint, #SearchCase decimal(38,0)
SELECT
#LastExponent = 79, -- 38 for bigint
#SearchCase = 729
;WITH CTE AS
(
SELECT
POWER(CAST(3 AS decimal(38,0)), ROW_NUMBER() OVER (ORDER BY c1.object_id)) AS Result,
ROW_NUMBER() OVER (ORDER BY c1.object_id) AS Exponent
FROM
sys.columns c1, sys.columns c2
)
SELECT
Result, Exponent
FROM
CTE
WHERE
Exponent <= #LastExponent
AND
Result = #SearchCase
With SET STATISTICS TIME ON it record the lowest possible, 1 millisecond.
Another approach is to generate a table on compile time. The good thing is, that you can extend this to powers of 4, 5, 6, 7, whatever
template<std::size_t... Is>
struct seq
{ };
template<std::size_t N, std::size_t... Is>
struct gen_seq : gen_seq<N-1, N-1, Is...>
{ };
template<std::size_t... Is>
struct gen_seq<0, Is...> : seq<Is...>
{ };
template<std::size_t N>
struct PowersOfThreeTable
{
std::size_t indexes[N];
std::size_t values[N];
static constexpr std::size_t size = N;
};
template<typename LambdaType, std::size_t... Is>
constexpr PowersOfThreeTable<sizeof...(Is)>
generatePowersOfThreeTable(seq<Is...>, LambdaType evalFunc)
{
return { {Is...}, {evalFunc(Is)...} };
}
template<std::size_t N, typename LambdaType>
constexpr PowersOfThreeTable<N> generatePowersOfThreeTable(LambdaType evalFunc)
{
return generatePowersOfThreeTable(gen_seq<N>(), evalFunc);
}
template<std::size_t Base, std::size_t Exp>
struct Pow
{
static constexpr std::size_t val = Base * Pow<Base, Exp-1ULL>::val;
};
template<std::size_t Base>
struct Pow<Base, 0ULL>
{
static constexpr std::size_t val = 1ULL;
};
template<std::size_t Base>
struct Pow<Base, 1ULL>
{
static constexpr std::size_t val = Base;
};
constexpr std::size_t tableFiller(std::size_t val)
{
return Pow<3ULL, val>::val;
}
bool isPowerOfThree(std::size_t N)
{
static constexpr unsigned tableSize = 41; //choosen by fair dice roll
static constexpr PowersOfThreeTable<tableSize> table =
generatePowersOfThreeTable<tableSize>(tableFiller);
for(auto a : table.values)
if(a == N)
return true;
return false;
}
I measured times (C#, Platform target x64) for some solutions.
using System;
class Program
{
static void Main()
{
var sw = System.Diagnostics.Stopwatch.StartNew();
for (uint n = ~0u; n > 0; n--) ;
Console.WriteLine(sw.Elapsed); // nada 1.1 s
sw.Restart();
for (uint n = ~0u; n > 0; n--) isPow3a(n);
Console.WriteLine(sw.Elapsed); // 3^20 17.3 s
sw.Restart();
for (uint n = ~0u; n > 0; n--) isPow3b(n);
Console.WriteLine(sw.Elapsed); // % / 10.6 s
Console.Read();
}
static bool isPow3a(uint n) // Elric
{
return n > 0 && 3486784401 % n == 0;
}
static bool isPow3b(uint n) // starblue
{
if (n > 0) while (n % 3 == 0) n /= 3;
return n == 1;
}
}
Another way (of splitting hairs).
using System;
class Program
{
static void Main()
{
Random rand = new Random(0); uint[] r = new uint[512];
for (int i = 0; i < 512; i++)
r[i] = (uint)(rand.Next(1 << 30)) << 2 | (uint)(rand.Next(4));
var sw = System.Diagnostics.Stopwatch.StartNew();
for (int i = 1 << 23; i > 0; i--)
for (int j = 0; j < 512; j++) ;
Console.WriteLine(sw.Elapsed); // 0.3 s
sw.Restart();
for (int i = 1 << 23; i > 0; i--)
for (int j = 0; j < 512; j++) isPow3c(r[j]);
Console.WriteLine(sw.Elapsed); // 10.6 s
sw.Restart();
for (int i = 1 << 23; i > 0; i--)
for (int j = 0; j < 512; j++) isPow3b(r[j]);
Console.WriteLine(sw.Elapsed); // 9.0 s
Console.Read();
}
static bool isPow3c(uint n)
{ return (n & 1) > 0 && 3486784401 % n == 0; }
static bool isPow3b(uint n)
{ if (n > 0) while (n % 3 == 0) n /= 3; return n == 1; }
}
Python program to check whether the number is a POWER of 3 or not.
def power(Num1):
while Num1 % 3 == 0:
Num1 /= 3
return Num1 == 1
Num1 = int(input("Enter a Number: "))
print(power(Num1))
Python solution
from math import floor
from math import log
def IsPowerOf3(number):
p = int(floor(log(number) / log(3)))
power_floor = pow(3, p)
power_ceil = power_floor * 3
if power_floor == number or power_ceil == number:
return True
return False
This is much faster than the simple divide by 3 solution.
Proof: 3 ^ p = number
p log(3) = log(number) (taking log both side)
p = log(number) / log(3)
Here's a general algorithm for finding out if a number is a power of another number:
bool IsPowerOf(int n,int b)
{
if (n > 1)
{
while (n % b == 0)
{
n /= b;
}
}
return n == 1;
}
#include<iostream>
#include<string>
#include<cmath>
using namespace std;
int main()
{
int n, power=0;
cout<<"enter a number"<<endl;
cin>>n;
if (n>0){
for(int i=0; i<=n; i++)
{
int r=n%3;
n=n/3;
if (r==0){
power++;
}
else{
cout<<"not exactly power of 3";
return 0;
}
}
}
cout<<"the power is "<<power<<endl;
}
This is a constant time method! Yes. O(1). For numbers of fixed length, say 32-bits.
Given that we need to check if an integer n is a power of 3, let us start thinking about this problem in terms of what information is already at hand.
1162261467 is the largest power of 3 that can fit into an Java int.
1162261467 = 3^19 + 0
The given n can be expressed as [(a power of 3) + (some x)]. I think it is fairly elementary to be able to prove that if x is 0(which happens iff n is a power of 3), 1162261467 % n = 0.
The general idea is that if X is some power of 3, X can be expressed as Y/3a, where a is some integer and X < Y. It follows the exact same principle for Y < X. The Y = X case is elementary.
So, to check if a given integer n is a power of three, check if n > 0 && 1162261467 % n == 0.
Python:
return n > 0 and 1162261467 % n == 0
OR Calculate log:
lg = round(log(n,3))
return 3**lg == n
1st approach is faster than the second one.